Computation of large-scale quadratic forms and transfer functions using the theory of moments, quadrature and Padé approximation

Abstract

Large-scale problems in scientific and engineering computing often require solutions involving large-scale matrices. In this paper, we survey numerical techniques for solving a variety of large-scale matrix computation problems, such as computing the entries and trace of the inverse of a matrix, computing the determinant of a matrix, and computing the transfer function of a linear dynamical system.

Most of these matrix computation problems can be cast as problems of computing quadratic forms uT ƒ(A)u involving a matrix functional ƒ(A). It can then be transformed into a Riemann-Stieltjes integral, which brings the theory of moments, orthogonal polynomials and the Lanczos process into the picture. For computing the transfer function, we focus on numerical techniques based on Padé approximation via the Lanczos process and the moment-matching property. We will also discuss issues related to the development of efficient numerical algorithms, including using Monte Carlo simulation.